Baby naps (almost 12 week old)

So my baby will be 3 months this month and I’m finding that he only sleeps for long stretches if he’s sleeping on me. He usually naps after a feed (and sometimes in between feeds if I cuddle/rock him; he can’t self-soothe)- he’s EBF - but if I try and transfer him to his carry cot, he’ll immediately wake up. I figured out that it’s because he can’t spread out like he can in his cot bed, which he’s used for a couple of weeks now after outgrowing his Moses basket, so should I do naps in there despite guidelines saying baby should always sleep in the same room as you? We have a baby monitor that has sound and video. The cot bed is in our room and obviously can’t be moved to wherever I am. Also, what do your babies nap in? Normal clothes and a blanket or a sleeping bag like a night? I’m worried if I put him in the sleeping bag he might get too hot?

Thank you in advance!